Successive minima

Question

Definition: The $n$ successive minima $\lambda_1,..,\lambda_n$ of $C$ with respect to lattice $L$ are defined as follow

$\lambda_i$ is the minimum of all positive reals $\lambda$ such that $\lambda C \cap L$ contains at least $i$ linear independent points. See also: https://en.wikipedia.org/wiki/Minkowski%27s_second_theorem

Determine the two successive minima of $$C=\{ (x_1,x_2) \in \mathbb{R}^2| \left| x_1- \sqrt{2}x_2 \right| \leq 1, \left| x_1- \sqrt{3}x_2 \right| \leq 1 \}$$ with respect to $\mathbb{Z}^2$.

By changing variables $u:= x_1-\sqrt{2}x_2$ and $v:=x_1 -\sqrt{3}x_2$ we have the square with side $2$. Hence, we have $$vol(C)= 2^2(\sqrt{2}+\sqrt{3})$$ By first Minkowski's convex body theorem, if we have $vol(\lambda C) \geq 2^2 $ then $\lambda C \cap \mathbb{Z}^2 \neq 0$. This implies $\lambda_1 \leq \sqrt{\dfrac{1}{\sqrt{3}+\sqrt{2}}}=\sqrt{\sqrt{3}-\sqrt{2}}$.

But I do not have lower bound for first successive minima. Does anyone have any idea?

Answer 1

For any positive $\lambda<1$, the set $\lambda C$ is a parallelogram containing a few integer points. For instance, for $\lambda=\frac 23$, it contains only the points $(0,0)$, $\pm (2,1)$, and $\pm (3,2)$, see the picture.

So for each $i\in\{1,2\}$ it is easy to find the minimum of all positive reals $\lambda$ such that $\lambda C \cap L$ contains at least $i$ linear independent points. Namely, for $i=2$ the points are $(0,0)$, $\pm (2,1)$, and $\pm (3,2)$, which implies $$\lambda=\max\{2-\sqrt{2},2-\sqrt{3},3-2\sqrt{2}, 2\sqrt{3}-3\}=2-\sqrt{2}=0.5857\dots.$$ For $i=1$ the points are $(0,0)$ and $\pm (3,2)$, which implies $$\lambda=\max\{3-2\sqrt{2}, 2\sqrt{3}-3\}=2\sqrt{3}-3=0.4641\dots.$$